Before you turn this problem in, make sure everything runs as expected. First, restart the kernel (in the menubar, select Kernel$\rightarrow$Restart) and then run all cells (in the menubar, select Cell$\rightarrow$Run All).

Make sure you fill in any place that says YOUR CODE HERE or "YOUR ANSWER HERE", as well as your name and collaborators below:

In [ ]:
NAME = "Ben Bitdiddle"
COLLABORATORS = "Alyssa P. Hacker"


For this problem set, we'll be using the Jupyter notebook:

Part A (2 points)¶

Write a function that returns a list of numbers, such that $x_i=i^2$, for $1\leq i \leq n$. Make sure it handles the case where $n<1$ by raising a ValueError.

In [ ]:
def squares(n):
"""Compute the squares of numbers from 1 to n, such that the
ith element of the returned list equals i^2.

"""
if n < 1:
raise ValueError
s = []
for i in range(n):
s.append(i**2)
return s


Your function should print [1, 4, 9, 16, 25, 36, 49, 64, 81, 100] for $n=10$. Check that it does:

In [ ]:
squares(10)

In [ ]:
# """Check that squares returns the correct output for several inputs"""
# assert squares(1) == [1]
# assert squares(2) == [1, 4]
# assert squares(10) == [1, 4, 9, 16, 25, 36, 49, 64, 81, 100]
# assert squares(11) == [1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121]

In [ ]:



Part B (1 point)¶

Using your squares function, write a function that computes the sum of the squares of the numbers from 1 to $n$. Your function should call the squares function -- it should NOT reimplement its functionality.

In [ ]:
def sum_of_squares(n):
"""Compute the sum of the squares of numbers from 1 to n."""
total = 0
s = squares(n)
for i in range(len(s)):
total += s[i]


The sum of squares from 1 to 10 should be 385. Verify that this is the answer you get:

In [ ]:
sum_of_squares(10)

In [ ]:
"""Check that sum_of_squares returns the correct answer for various inputs."""
assert sum_of_squares(1) == 1
assert sum_of_squares(2) == 5
assert sum_of_squares(10) == 385
assert sum_of_squares(11) == 506

In [ ]:
"""Check that sum_of_squares relies on squares."""
orig_squares = squares
del squares
try:
sum_of_squares(1)
except NameError:
pass
else:
raise AssertionError("sum_of_squares does not use squares")
finally:
squares = orig_squares


Part C (1 point)¶

Using LaTeX math notation, write out the equation that is implemented by your sum_of_squares function.

$\sum_{i=0}^n i^2$

Part D (2 points)¶

Find a usecase for your sum_of_squares function and implement that usecase in the cell below.

In [ ]:
# YOUR CODE HERE
raise NotImplementedError()


Part E (4 points)¶

State the formulae for an arithmetic and geometric sum and verify them numerically for an example of your choice.

$\sum x^i = \frac{1}{1-x}$